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    "# 天体运动的物理公式\n",
    "\n",
    "## 1. 开普勒三定律\n",
    "\n",
    "### 第一定律（轨道定律）\n",
    "所有行星绕太阳运动的轨道都是椭圆，太阳处在椭圆的一个焦点上。\n",
    "\n",
    "**椭圆轨道方程：**\n",
    "$$ \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 $$\n",
    "其中 $a$ 为半长轴，$b$ 为半短轴\n",
    "\n",
    "### 第二定律（面积定律）\n",
    "对任意一个行星来说，它与太阳的连线在相等的时间内扫过相等的面积。\n",
    "\n",
    "**面积速度：**\n",
    "$$ \\frac{dA}{dt} = \\frac{1}{2}r^2\\frac{d\\theta}{dt} = {常数} $$\n",
    "\n",
    "### 第三定律（周期定律）\n",
    "所有行星的轨道半长轴的三次方跟公转周期的二次方的比值都相等。\n",
    "\n",
    "**数学表达式：**\n",
    "$$ \\frac{a^3}{T^2} = k $$\n",
    "对于太阳系：$ \\frac{a^3}{T^2} = \\frac{GM}{4\\pi^2} $\n",
    "\n",
    "## 2. 万有引力定律\n",
    "\n",
    "### 万有引力公式\n",
    "$$ F = G\\frac{Mm}{r^2} $$\n",
    "\n",
    "**其中：**\n",
    "- $G = 6.67430 \\times 10^{-11} {N·m}^2/{kg}^2$（引力常数）\n",
    "- $M$：中心天体质量\n",
    "- $m$：环绕天体质量\n",
    "- $r$：两天体中心距离\n",
    "\n",
    "## 3. 圆周轨道运动\n",
    "\n",
    "### 轨道速度\n",
    "$$ v = \\sqrt{\\frac{GM}{r}} $$\n",
    "\n",
    "### 轨道周期\n",
    "$$ T = 2\\pi\\sqrt{\\frac{r^3}{GM}} $$\n",
    "\n",
    "### 角速度\n",
    "$$ \\omega = \\sqrt{\\frac{GM}{r^3}} $$\n",
    "\n",
    "### 向心加速度\n",
    "$$ a = \\frac{GM}{r^2} = \\frac{v^2}{r} $$\n",
    "\n",
    "## 4. 椭圆轨道运动\n",
    "\n",
    "### 能量守恒\n",
    "$$ E = \\frac{1}{2}mv^2 - G\\frac{Mm}{r} = -\\frac{GMm}{2a} $$\n",
    "\n",
    "### 速度公式（vis-viva方程）\n",
    "$$ v^2 = GM\\left(\\frac{2}{r} - \\frac{1}{a}\\right) $$\n",
    "\n",
    "### 近日点和远日点速度\n",
    "- **近日点**：$ v_p = \\sqrt{\\frac{GM}{a}\\cdot\\frac{1+e}{1-e}} $\n",
    "- **远日点**：$ v_a = \\sqrt{\\frac{GM}{a}\\cdot\\frac{1-e}{1+e}} $\n",
    "\n",
    "## 5. 宇宙速度\n",
    "\n",
    "### 第一宇宙速度（环绕速度）\n",
    "$$ v_1 = \\sqrt{\\frac{GM}{R}} \\approx 7.9 { km/s} $$\n",
    "\n",
    "### 第二宇宙速度（逃逸速度）\n",
    "$$ v_2 = \\sqrt{\\frac{2GM}{R}} \\approx 11.2 { km/s} $$\n",
    "\n",
    "### 第三宇宙速度（太阳系逃逸速度）\n",
    "$$ v_3 \\approx 16.7 { km/s} $$\n",
    "\n",
    "## 6. 引力势能\n",
    "\n",
    "### 引力势能\n",
    "$$ U = -G\\frac{Mm}{r} $$\n",
    "\n",
    "### 机械能\n",
    "$$ E = \\frac{1}{2}mv^2 - G\\frac{Mm}{r} $$\n",
    "\n",
    "## 7. 多体问题近似\n",
    "\n",
    "### 约化质量\n",
    "$$ \\mu = \\frac{m_1 m_2}{m_1 + m_2} $$\n",
    "\n",
    "### 相对运动方程\n",
    "$$ \\mu\\frac{d^2\\mathbf{r}}{dt^2} = -G\\frac{m_1 m_2}{r^2}\\hat{\\mathbf{r}} $$\n",
    "\n",
    "## 8. 潮汐力\n",
    "\n",
    "### 潮汐加速度\n",
    "$$ a_{{tidal}} = \\frac{2GMr}{d^3} $$\n",
    "\n",
    "**其中：**\n",
    "- $M$：引起潮汐的天体质量\n",
    "- $r$：被测天体的半径\n",
    "- $d$：两天体中心距离\n",
    "\n",
    "## 9. 特殊轨道\n",
    "\n",
    "### 同步轨道\n",
    "$$ r_{{sync}} = \\sqrt[3]{\\frac{GMT^2}{4\\pi^2}} $$\n",
    "\n",
    "### 拉格朗日点\n",
    "五个特殊平衡点（L₁-L₅），其中L₁、L₂、L₅常用于航天任务\n",
    "\n",
    "---\n",
    "\n",
    "## 📊 天体参数（太阳系示例）\n",
    "\n",
    "| 天体 | 质量 (kg) | 半径 (km) | 逃逸速度 (km/s) |\n",
    "|------|-----------|-----------|-----------------|\n",
    "| 太阳 | $1.989 \\times 10^{30}$ | 696,340 | 617.7 |\n",
    "| 地球 | $5.972 \\times 10^{24}$ | 6,371 | 11.2 |\n",
    "| 月球 | $7.342 \\times 10^{22}$ | 1,737 | 2.4 |\n",
    "\n",
    "## 💡 重要概念\n",
    "\n",
    "1. **开普勒定律**：描述行星运动的基本规律\n",
    "2. **万有引力**：天体间相互作用的基本力\n",
    "3. **轨道能量**：负值表示束缚轨道，零值表示抛物线轨道，正值表示双曲线轨道\n",
    "4. **宇宙速度**：航天器需要达到的最小速度\n",
    "\n",
    "> 这些公式是理解天体运动的基础，广泛应用于天文学、航天工程和天体物理学领域。\n",
    "\n",
    "## 10. 相对论修正\n",
    "\n",
    "### 近日点进动\n",
    "$$ \\Delta\\theta = \\frac{6\\pi GM}{c^2 a(1-e^2)} { rad/轨道} $$\n",
    "\n",
    "### 引力红移\n",
    "$$ \\frac{\\Delta f}{f} = \\frac{GM}{c^2 r} $$\n",
    "\n",
    "**其中 $c$ 为光速**"
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    "# 天体椭圆轨道中速度与周期的关系\n",
    "\n",
    "## 1. 开普勒第三定律（周期定律）\n",
    "\n",
    "### 数学表达式\n",
    "$$ \\frac{a^3}{T^2} = \\frac{GM}{4\\pi^2} $$\n",
    "\n",
    "### 改写形式\n",
    "$$ T = 2\\pi\\sqrt{\\frac{a^3}{GM}} $$\n",
    "\n",
    "**其中：**\n",
    "- $T$：公转周期\n",
    "- $a$：轨道半长轴\n",
    "- $G$：万有引力常量（$6.67 \\times 10^{-11}$ N·m²/kg²）\n",
    "- $M$：中心天体质量\n",
    "\n",
    "## 2. 椭圆轨道速度公式（vis-viva方程）\n",
    "\n",
    "### 通用速度公式\n",
    "$$ v^2 = GM\\left(\\frac{2}{r} - \\frac{1}{a}\\right) $$\n",
    "\n",
    "**其中：**\n",
    "- $v$：天体在距离$r$处的速度\n",
    "- $r$：瞬时距离（从焦点到天体的距离）\n",
    "\n",
    "## 3. 特殊位置的速度\n",
    "\n",
    "### 近日点速度（最大速度）\n",
    "$$ v_p = \\sqrt{\\frac{GM}{a} \\cdot \\frac{1+e}{1-e}} $$\n",
    "\n",
    "### 远日点速度（最小速度）\n",
    "$$ v_a = \\sqrt{\\frac{GM}{a} \\cdot \\frac{1-e}{1+e}} $$\n",
    "\n",
    "**其中：**\n",
    "- $e$：轨道偏心率（$0 \\leq e < 1$）\n",
    "\n",
    "## 4. 速度与周期的关系推导\n",
    "\n",
    "### 由开普勒第二定律（面积定律）\n",
    "$$ \\frac{dA}{dt} = \\frac{1}{2}r^2\\frac{d\\theta}{dt} = {常数} $$\n",
    "\n",
    "### 轨道周长与周期\n",
    "一个周期内扫过的总面积：\n",
    "$$ A = \\pi ab = \\pi a^2\\sqrt{1-e^2} $$\n",
    "\n",
    "### 平均角速度\n",
    "$$ \\bar{\\omega} = \\frac{2\\pi}{T} $$\n",
    "\n",
    "### 平均速度\n",
    "$$ \\bar{v} = \\frac{2\\pi a}{T}\\sqrt{1-e^2} $$\n",
    "\n",
    "## 5. 重要关系式\n",
    "\n",
    "### 速度极值比\n",
    "$$ \\frac{v_p}{v_a} = \\frac{1+e}{1-e} $$\n",
    "\n",
    "### 半长轴与周期关系\n",
    "$$ a = \\left(\\frac{GMT^2}{4\\pi^2}\\right)^{1/3} $$\n",
    "\n",
    "### 圆轨道特例（$e=0$）\n",
    "$$ v = \\sqrt{\\frac{GM}{a}} $$\n",
    "$$ T = 2\\pi\\sqrt{\\frac{a^3}{GM}} $$\n",
    "\n",
    "## 6. 应用示例\n",
    "\n",
    "### 地球绕太阳运动\n",
    "- 半长轴：$a = 1.496 \\times 10^{11}$ m\n",
    "- 太阳质量：$M = 1.989 \\times 10^{30}$ kg\n",
    "- 偏心率：$e = 0.0167$\n",
    "\n",
    "**计算：**\n",
    "$$ T = 2\\pi\\sqrt{\\frac{(1.496\\times10^{11})^3}{6.67\\times10^{-11} \\times 1.989\\times10^{30}}} \\approx 3.156\\times10^7 { s} $$\n",
    "\n",
    "$$ v_p = \\sqrt{\\frac{6.67\\times10^{-11} \\times 1.989\\times10^{30}}{1.496\\times10^{11}} \\cdot \\frac{1+0.0167}{1-0.0167}} \\approx 30.3 { km/s} $$\n",
    "\n",
    "## 7. 物理意义\n",
    "\n",
    "1. **能量守恒**：vis-viva方程体现了机械能守恒\n",
    "2. **角动量守恒**：开普勒第二定律反映了角动量守恒\n",
    "3. **轨道参数相关性**：速度分布由$a$和$e$共同决定\n",
    "4. **周期确定性**：周期只与半长轴有关，与偏心率无关\n",
    "\n",
    "## 8. 记忆技巧\n",
    "\n",
    "- **周期**：只与半长轴有关，$T \\propto a^{3/2}$\n",
    "- **速度**：与距离和半长轴都有关，$v \\propto \\sqrt{\\frac{2}{r} - \\frac{1}{a}}$\n",
    "- **极值速度**：$v_p > v_a$，比值取决于偏心率\n",
    "\n",
    "> 这些关系是天体力学的基础，适用于所有人造卫星和自然天体的椭圆轨道运动。"
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